Resampling methods for improved wavelet-based multiple hypothesis testing of parametric maps in functional MRI

Two- or three-dimensional wavelet transforms have been considered as a basis for multiple hypothesis testing of parametric maps derived from functional magnetic resonance imaging (fMRI) experiments. Most of the previous approaches have assumed that the noise variance is equally distributed across levels of the transform. Here we show that this assumption is unrealistic; fMRI parameter maps typically have more similarity to a 1/f-type spatial covariance with greater variance in 2D wavelet coefficients representing lower spatial frequencies, or coarser spatial features, in the maps. To address this issue we resample the fMRI time series data in the wavelet domain (using a 1D discrete wavelet transform [DWT]) to produce a set of permuted parametric maps that are decomposed (using a 2D DWT) to estimate level-specific variances of the 2D wavelet coefficients under the null hypothesis. These resampling-based estimates of the "wavelet variance spectrum" are substituted in a Bayesian bivariate shrinkage operator to denoise the observed 2D wavelet coefficients, which are then inverted to reconstitute the observed, denoised map in the spatial domain. Multiple hypothesis testing controlling the false discovery rate in the observed, denoised maps then proceeds in the spatial domain, using thresholds derived from an independent set of permuted, denoised maps. We show empirically that this more realistic, resampling-based algorithm for wavelet-based denoising and multiple hypothesis testing has good Type I error control and can detect experimentally engendered signals in data acquired during auditory-linguistic processing.